Assume we have a finite irreducible aperiodic Markov Chain represented by the matrix $A$ and we take the $i$-th vertex out (i.e. remove the corresponding column and row) obtaining $A_{-i}$.
Question: Is there much* that can be said about the largest eigenvalue $\lambda_{max}(A_{-i})$ of $A_{-i}$, assuming the graph of $A_{-i}$ is connected, apart from $0<\lambda_{max}(A_{-i})<1$ ?
What about $\sum_j\lambda_{max}(A_{-j})$?
*as in estimates, or explicitly as a function of the largest eigenvalue of $A$, its associated eigenvector and the deleted column and row.